# Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics

Xavier Blanc; Claude Le Bris; Frédéric Legoll

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 39, Issue: 4, page 797-826
- ISSN: 0764-583X

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topBlanc, Xavier, Le Bris, Claude, and Legoll, Frédéric. "Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics." ESAIM: Mathematical Modelling and Numerical Analysis 39.4 (2010): 797-826. <http://eudml.org/doc/194287>.

@article{Blanc2010,

abstract = {
In order to describe a solid which deforms smoothly in some region, but
non smoothly in some other region, many multiscale methods have recently
been proposed. They aim at coupling an atomistic model (discrete
mechanics) with a macroscopic model
(continuum mechanics).
We provide here a theoretical ground for such a coupling in a
one-dimensional setting. We briefly study the general case of a convex
energy, and next concentrate on
a specific example of a nonconvex energy, the Lennard-Jones case. In the
latter situation, we prove that the discretization needs to account in
an adequate way for the coexistence of a discrete model and a continuous
one. Otherwise, spurious discretization effects may appear.
We provide a numerical analysis of the approach.
},

author = {Blanc, Xavier, Le Bris, Claude, Legoll, Frédéric},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Multiscale methods; variational problems; continuum mechanics;
discrete mechanics.; discrete mechanics},

language = {eng},

month = {3},

number = {4},

pages = {797-826},

publisher = {EDP Sciences},

title = {Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics},

url = {http://eudml.org/doc/194287},

volume = {39},

year = {2010},

}

TY - JOUR

AU - Blanc, Xavier

AU - Le Bris, Claude

AU - Legoll, Frédéric

TI - Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 39

IS - 4

SP - 797

EP - 826

AB -
In order to describe a solid which deforms smoothly in some region, but
non smoothly in some other region, many multiscale methods have recently
been proposed. They aim at coupling an atomistic model (discrete
mechanics) with a macroscopic model
(continuum mechanics).
We provide here a theoretical ground for such a coupling in a
one-dimensional setting. We briefly study the general case of a convex
energy, and next concentrate on
a specific example of a nonconvex energy, the Lennard-Jones case. In the
latter situation, we prove that the discretization needs to account in
an adequate way for the coexistence of a discrete model and a continuous
one. Otherwise, spurious discretization effects may appear.
We provide a numerical analysis of the approach.

LA - eng

KW - Multiscale methods; variational problems; continuum mechanics;
discrete mechanics.; discrete mechanics

UR - http://eudml.org/doc/194287

ER -

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